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arXiv · 2404.10905

Intermediate Domains for Scalar Conservation Laws

Fabio Ancona·Alberto Bressan·Elio Marconi·Luca Talamini
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Abstract

For a scalar conservation law with strictly convex flux, by Oleinik's estimates the total variation of a solution with initial data $\overline{u}\in \bf{L}^\infty(\mathbb R)$ decays like $t^{-1}$. This paper introduces a class of intermediate domains $\mathcal P_\alpha$, $0<\alpha<1$, such that for $\overline u\in \mathcal P_\alpha$ a faster decay rate is achieved: $\mathrm{Tot.Var.}\bigl\{ u(t,\cdot)\bigr\}\sim t^{\alpha-1}$. A key ingredient of the analysis is a ``Fourier-type" decomposition of $\overline u$ into components which oscillate more and more rapidly. The results aim at extending the theory of fractional domains for analytic semigroups to an entirely nonlinear setting.

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Fabio Ancona, Alberto Bressan, Elio Marconi, Luca Talamini. 2024-04-16. Intermediate Domains for Scalar Conservation Laws. https://arxiv.org/abs/2404.10905

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